(a) Assume the equation x = At 3 + Bt describes the motion of a particular object, with x having the dimension of length and t having the dimension of time. Determine the dimensions of the constants A and B . (b) Determine the dimensions of the derivative dx / dt = 3 At 2 + B .

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(a) To determine The dimensions of the constant A and B in the equation x = A t 3 + B t . Explanation Write the equation of the motion of a particular object having x as function of t x = A t 3 + B t (I) Here, x is the position of the object, t is the time, A is any constant and B is any constant. The dimension of x is L and the dimension of t is T . Assume the dimension of A is L a T b and the dimension of B is L c T d . Since the given equation is dimensionally correct. So, the dimension on left side must balance the dimension of right side. Equating the dimension of x and A t 3 . Substitute L for x and L a T b for A t 3 . L = ( L a T b ) T 3 L = L a ⋅ T b + 3 Equating the exponent on both side of the equation, a = 1 b + 3 = 0 b = − 3 Substitute 1 for a and − 3 for b to find the dimension of A (b) To determine The dimensions of the derivative d x d t in the equation d x d t = 3 A t 2 + B .

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Physics for Scientists and Enginee...

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Raymond A. Serway + 1 other

Publisher: Cengage Learning

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Physics for Scientists and Enginee...

10th Edition

Raymond A. Serway + 1 other

Publisher: Cengage Learning

ISBN: 9781337553278

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Answer 6

(a)

To determine

The dimensions of the constant A and B in the equation x=At3+Bt.

Explanation of Solution

Write the equation of the motion of a particular object having x as function of t

x=At3+Bt (I)

Here, x is the position of the object, t is the time, A is any constant and B is any constant.

The dimension of x is L and the dimension of t is T.

Assume the dimension of A is LaTb and the dimension of B is LcTd.

Since the given equation is dimensionally correct. So, the dimension on left side must balance the dimension of right side.

Equating the dimension of x and At3.

Substitute L for x and LaTb for At3.

L=(LaTb)T3L=La⋅Tb+3

Equating the exponent on both side of the equation,

a=1

b+3=0b=−3

Substitute 1 for a and −3 for b to find the dimension of A

Answer 7

(b)

To determine

The dimensions of the derivative dxdt in the equation dxdt=3At2+B.

(a) Assume the equation x = At 3 + Bt describes the motion of a particular object, with x having the dimension of length and t having the dimension of time. Determine the dimensions of the constants A and B . (b) Determine the dimensions of the derivative dx / dt = 3 At 2 + B .

Physics for Scientists and Enginee...

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Physics for Scientists and Enginee...

(a) Assume the equation x = At³ + Bt describes the motion of a particular object, with x having the dimension of length and t having the dimension of time. Determine the dimensions of the constants A and B. (b) Determine the dimensions of the derivative dx/dt = 3At² + B.

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(a) Assume the equation x = At 3 + Bt describes the motion of a particular object, with x having the dimension of length and t having the dimension of time. Determine the dimensions of the constants A and B . (b) Determine the dimensions of the derivative dx / dt = 3 At 2 + B .

Physics for Scientists and Enginee...

Solutions

Physics for Scientists and Enginee...